First-Principles Calculation of the Elastic Moduli of Sheet Silicates and Their Application to Shale Anisotropy

American Mineralogist, Volume 96, pages 125–137, 2011 First-principles calculation of the elastic moduli of sheet silicates and their application to shale anisotropy B. MILITZER,1,2,* H.-R. WENK ,1 S. STACK H OU S E ,1 AND L. STIXRUDE 3 1Department of Earth and Planetary Science, University of California, Berkeley, California 94720, U.S.A. 2Department of Astronomy, University of California, Berkeley, California 94720, U.S.A. 3Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, U.K. Abs TRACT The full elastic tensors of the sheet silicates muscovite, illite-smectite, kaolinite, dickite, and nacrite have been derived with first-principles calculations based on density functional theory. For muscovite, there is excellent agreement between calculated properties and experimental results. The influence of cation disorder was investigated and found to be minimal. On the other hand, stacking disorder is found to be of some relevance for kaolin minerals. The corresponding single-crystal seismic wave velocities were also derived for each phase. These revealed that kaolin minerals exhibit a distinct type of seismic anisotropy, which we relate to hydrogen bonding. The elastic properties of a shale aggregate was predicted by averaging the calculated properties of the contributing mineral phases over their orientation distributions. Calculated elastic properties display higher stiffness and lower p-wave anisotropy. The difference is likely due to the presence of oriented flattened pores in natural samples that are not taken into account in the averaging. Keywords: Elasticity, clay, ab initio calculations, sheet silicates, seismic anisotropy INTRODUCTION the case of shales. One reason is that porosity and grain contacts Sheet silicates are among the minerals with highest elastic are inadequately accounted for. Another reason is the uncertainty anisotropy. Aggregates containing oriented sheet silicates, such in the single-crystal elastic properties of sheet silicates (Chen as schists and shales, also display very high anisotropies and and Evans 2006). this has important implications for interpreting seismic travel Despite their importance, the single-crystal elastic moduli of times through such rocks. Shales are the most abundant rocks in clay minerals have not been measured experimentally, because of sedimentary basins and relevant for petroleum deposits (e.g., Jo- technical difficulties associated with the small grain size. There hansen et al. 2004) as well as for carbon sequestration (Chadwick is also considerable uncertainty in atomic force microscopy et al. 2004). Their physical properties are of great importance (Prasad et al. 2002). For illite, elastic constants of the analog in exploration geophysics (e.g., Mavko et al. 1998; Wang et al. mineral, muscovite, have traditionally been used (Vaughan and 2001). Elastic properties of shales have been measured with Guggenheim 1986; Zhang et al. 2009) but considerable uncer- ultrasound techniques (e.g., Hornby 1998; Hornby et al. 1994; tainty arises from using such analogies. For kaolinite, values Johnston and Christensen 1995), because of their importance for from first-principles calculations exist for the ideal structure seismic prospecting of hydrocarbon deposits (e.g., Banik 1984). (Sato et al. 2005; Mercier and Le Page 2008). However, the In metals and igneous and metamorphic rocks, anisotropic ag- structure of kaolin minerals in natural samples is expected to gregate properties can be satisfactorily predicted by averaging the differ considerably from the ideal kaolinite. Moreover, other single-crystal elastic properties over the orientation distribution equally important sheet silicates have not yet been examined (Bunge 1985; Barruol and Kern 1996; Mainprice and Humbert by experiment or theory, including illite-smectite. In view of 1994). In the case of shale, anisotropy can be calculated with this, in the present work, we have calculated the single-crystal this method, but absolute values of predicted elastic stiffness elastic properties of illite-smectite and the kaolin minerals kao- coefficients are over a factor of two higher than the measured linite, dickite, and nacrite, which have structures more similar results (Valcke et al. 2006; Voltolini et al. 2009; Wenk et al. 2008). to those found in nature. In addition, to establish and test our Currently only empirical models are available to describe the method, we have computed the elastic properties of muscovite, elasticity of shales that do not rely on measured microstructural for which analogous experimental data exists. The computed properties (e.g., Bayuk et al. 2007; Ougier-Simoni et al. 2008; elastic properties are then used to predict the elastic properties Ponte Castaneda and Willis 1995; Sayers 1994). There are vari- of the classic Kimmeridge Clay shale (age 151–156 Ma) from ous reasons why the simple averaging schemes are limited in Hornby (1998) for which experimental elastic properties are available, as well as orientation distributions of component * E-mail: [email protected] phases (Wenk et al. 2010). 0003-004X/11/0001–125$05.00/DOI: 10.2138/am.2011.3558 125 126 MILITZER ET AL.: AB INITIO CalculatioN OF ELASTIC MODULI OF SHEET Silicates → → → Wenk 2009) with an orthogonal coordinate system defined by Z||→c, Y||→c×a→, and X CO M PUTATIONAL M ET H OD S → → → = Y×Z and (2) a setting where the Z is perpendicular to the tetrahedral planes (001) We used the Vienna ab initio simulation package (VASP) (Kresse and Hafner → → → → → → → → in both monoclinic and triclinic crystal structures: X||a, Z||a×b, and Y = Z×X. The 1993; Kresse and Furthmüller 1996) to perform density functional calculations latter choice preserves the direction of C33 when comparing different stackings in using either the local density approximation (LDA) or the → Perdew-Burke-Ernzerhof kaolinite polytypes that have different c axes. (Perdew et al. 1996) generalized gradient approximation (GGA) in combination with the projector augmented-wave method (Blöchl 1994). The wave functions of the valence electrons were expanded in a plane-wave basis with an energy cut-off MODEL S Y S TE ms of 586 eV or higher and the Brillouin zone was sampled using 4 × 2 × 2 and 6 × 4 × 4 k-points grids (Monkhorst and Pack 1976) depending on the size of the unit Muscovite –3 cell. All structures were relaxed until all forces were <10 eV/A. Muscovite KAl2[AlSi3O10](OH)2 is a 2:1 sheet silicate, with To calculate the elastic constants we used the finite strain method (Karki et al. → ↔ each layer comprising an octahedral sheet fused between two 1997, 1997b, 2001). The lattice vectors of the strained unit cell, (a→′, b′, →c′) ≡ A′, → ↔ ↔ ↔ ↔ ↔ were obtained from the unstrained cell, (a→, b, →c) ≡ A, using A′ = (1 + ε) A, where tetrahedral sheets (Fig. 1). Due to isomorphic substitution, it ↔ ↔ 1 is the unit matrix and ε is the strain tensor. The diagonal and off-diagonal strain exhibits structural disorder in the tetrahedral sheets (Collins and tensors were of the form (Ravindran et al. 1998) Catlow 1992), with 25% of sites occupied by Al3+ cations and 75% by Si4+ cations. This leads to the layers having a net nega- δ 00 000 tive charge, which is balanced by K+ anions residing between ; , ε = 1 000 ε4 = 00δ / 2 the layers. 000 02δ / 0 Since it is possible that cation disorder will affect the elastic properties of the phase, we performed first-principles calculations where the indices are given in Voigt notation. The remaining strain tensors are to determine the most favorable arrangements. obtained by index permutations. For each strain, we performed two calculations 4+ 3+ using δ = ± 0.005 and determined the corresponding stresses. The elastic constants Isomorphic substitution of Si by Al , in the tetrahedral 3+ were then determined from simple stress-strain relations. Consistent results were sheets, creates a local negative charge in the region of the Al also obtained for other values of δ. Mainprice et al. (2008) performed similar cation. It is therefore reasonable to assume that Al3+ cations calculations for talc. within the same tetrahedral layer, will tend to repel one another. The error associated with our computed elastic constants, due to controlled Our calculations support this. For an 84 atom unit cell, placing ± approximations, is estimated to be < 2 GPa, with the major part of this associated 3+ with the careful, but slightly imperfect, structural relaxation and use of a finite two Al cations in the same tetrahedral sheet increases the energy 3+ strain. In addition to this, we must also take into account the uncontrolled ap- by 1–2 eV, while placing all four Al cations in the same sheet proximation of adopting a specific exchange-correlation functional. While LDA increases it by 7 eV. Calculations for heterogeneous distributions and GGA have been shown to predict many properties of a variety of materials in larger super cells showed a similar trend. This provides a strong with remarkable accuracy, neither approach is perfect. LDA tends to overbind the 3+ P=0 argument for having similar Al cation concentrations in each sheets in sheet silicates, and zero-pressure densities predicted with LDA (ρLDA) are too high (by 3% for muscovite). In contrast, GGA tends to underbind the layer, to reduce unfavorable interactions between them. P=0 sheets, and zero-pressure densities predicted with GGA (ρGGA) are too low (by If one considers the conventional unit cell, assuming an equal 5% for muscovite). Since the computed elastic constants are affected by changes Al3+ concentration in each layer, there is only one Al3+ cation to in density, we can improve the accuracy of the predicted constants by adopting distribute among four sites. Every site is energetically equivalent ρP=0 the experimental density ( EXP) and relaxing the structure at fixed volume. In this 4+ 3+ approach it is important to account for uncertainties in the volume, structure, and and leads to the formation of a Si -Al chain parallel to the b composition of the experimental sample and differences between it and the model lattice vector (Fig.

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